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Vapour phase Aqueous solution
Hydrocarbon Aqueous solution
Vapour phase Aqueous solution
Hydrophobic group
Hydrophilic group
Surfactant molecules accumulate in surface layer
Micelles form at
/critical micelle concentration
Figure 6.1 Orientation of amphiphiles at (a) solution/ vapour interface, and (b) hydrocarbon/solution interface.
Log c
Figure 6.2 Typical plot of the surface tension, y, against logarithm of surfactant concentration, c.
exerted between two water molecules, the contracting power of the surface is reduced and so therefore is the surface tension. In some cases the interfacial tension between two liquids may be reduced to such a low level (10 3 mN m01) that spontaneous emulsification of the two immiscible liquids is observed. These very low interfacial tensions are of relevance in understanding the formation and stabilisation of emulsions and are dealt with in more detail in Chapter 7.
6.2.2 Change of surface tension with surfactant concentration  the critical micelle concentration
Fig. 6.2 shows a typical plot of surface tension against the logarithm of concentration for a surfactant solution. Appreciable lowering of surface tension is evident even at low concentrations. As the surfactant concentration is increased, the surface tension continues to decrease as the concentration of surfactant molecules at the surface increases. A concentration is reached, however, when the surface layer becomes saturated with surfactant molecules and no further decrease in surface tension is possible. An alternative means of shielding the hydrophobic portion of the amphiphile from the aqueous environment now occurs as the surfactant molecules form small spherical aggregates or micelles in the bulk of the solution. The hydrophobic groups of the surfactants form the core of these aggregates and are protected from contact with water by their hydrophilic groups, which form a shell around them. The concentration of surfactant molecules in the surface layer remains approximately constant in the presence of micelles and hence the ylog concentration plot becomes almost horizontal. The concentration at which the micelles first form in solution is called the critical micelle concentration (cmc) and corresponds to the concentration at which there is an abrupt change of slope of the plot. We will consider the formation and properties of micelles in more detail in section 6.3. At the moment we will concentrate on the region of the ylog concentration plot below the cmc and see how it is possible to calculate the area occupied by a surfactant molecule at the surface using the Gibbs equation.
6.2.3 Gibbs adsorption equation
It is important to remember that an equilibrium is established between the surfactant molecules at the surface or interface and those remaining in the bulk of the solution. This equilibrium is expressed in terms of the Gibbs equation. In developing this expression it is necessary to imagine a definite boundary between the bulk of the solution and the interfacial layer (see Fig. 6.3). The real system containing the interfacial layer is then compared with this reference system, in which
Box 6.2 The Gibbs equation
We can treat the thermodynamics of the surface layer in a similar way to the bulk of the solution. The energy change, dU, accompanying an infinitesimal, reversible change in the system is given by dU = dqrov  dw dU=TdSdv
where dqrev and dw are, respectively, the heat absorbed and the work done during the reversible change (see section 3.1).
For an open system (one in which there is transfer of material between phases) equation (6.2) must be written dU = T dS  dw+ E a, dn, (6.3)
where ¡ui and n; are the chemical potential and number of moles respectively of the ith component.
When applying equation (6.3) to the surface layer, the work is that required to increase the area of the surface by an infinitesimal amount, dA, at constant T, P and n. This work is done against the surface tension and is given by equation (6.1) as d w = y dA.
Thus, equation (6.3) becomes dUs = TdSs + y dA + E a, dns (6.4)
where the superscript, s, denotes the surface layer.
If the energy, entropy and number of moles of component are allowed to increase from zero to some finite value, equation (6.4) becomes
Us = TSS + yA + E a,ns (6.5) General differentiation of equation (6.5) gives dUs = TdSs + Ss dT+y dA + A dy
Comparison with equation (6.4) gives
At constant temperature equation (6.7) becomes dy = E r , da i (6.8)
where r, = nS/A and is termed the surface excess concentration. r,. is the amount of the ith component in the surface phase s, in excess of that which there would have been had the bulk phases a and b extended to the dividing surface without change in composition.
For a twocomponent system at constant temperature, equation (6.8) reduces to dy = r da,  r2 da2 (6.9)
where subscripts 1 and 2 denote solvent and solute, respectively. The surface excess concentrations are defined relative to an arbitrarily chosen dividing surface. A convenient choice of location of this surface is that at which the surface excess concentration of the solvent, r,, is zero. Indeed, this is the most realistic position since we are now considering the surface layer of adsorbed solute. Equation (6.9) then becomes dy = r2 da2 (6.10)
The chemical potential of the solute is given by equation (3.52) as a2 =a + RT In a2
Substituting in equation (6.10) gives the Gibbs equation:
a2 dy
RT da2

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